2012-47.md

course	course_year	question_number	tags	title	year
Methods	IB	47	IB 2012 Methods	Paper 2, Section I, C	2012

Using the method of characteristics, obtain a solution to the equation

$$u_{x}+2 x u_{y}=y$$

subject to the Cauchy data $u(0, y)=1+y^{2}$ for $-\frac{1}{2}<y<\frac{1}{2}$.

Sketch the characteristics and specify the greatest region of the plane in which a unique solution exists.